I'm starting to studying Control theory of partial differential equations. Until now i have only studied systems on finite dimensional spaces and some particular equations(wave equation, heat equation, and others). So i'm interested in study the theory in infinite dimensional spaces, so i would like to know a general (detailed if it is possible) idea on what happen in this case, the relation between controllability and observability and if it is possible a briefly explication of the Hilbert Uniqueness Method.
Any reference would be helpful, and thank you in advance.
I suggest the book of Jerzy Zabczyk https://www.springer.com/gb/book/9780817647322 You will find all the details you need in chapter 4. For the HUM I suggest the first volume of Jaque Luis Lions https://books.google.dz/books?id=ZoW6XwAACAAJ&dq=exact+controllability+lions&hl=fr&sa=X&ved=0ahUKEwiC3pK2wKTbAhUB6qQKHYmZAmYQ6AEIKTAA.